5.1 midsegment theorem and coordinate proof
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5.1 Midsegment Theorem and Coordinate Proof. Essential Question:. How do you write a coordinate proof?. You will use properties of midsegments and write coordinate proofs. - PowerPoint PPT PresentationTRANSCRIPT
5.1 Midsegment Theorem and Coordinate Proof
• You will use properties of midsegments and write coordinate proofs.
Essential Question:
• How do you write a coordinate proof?
You will learn how to answer this question by placing a figure in the coordinate plane, assigning coordinates to the vertices, and then using the midpoint, distance, and/or slope formulas.
Warm-Up ExercisesEXAMPLE 1 Use the Midsegment Theorem to find lengths
CONSTRUCTION
SOLUTION
UV =12 RT =
12 ( 90 in.) = 45 in.
RS = 2 VW = 2 ( 57 in.) = 114 in.
Triangles are used for strength in roof trusses. In the diagram, UV and VW are midsegments of
Find UV and RS.RST.
Warm-Up ExercisesGUIDED PRACTICE for Example 1
1. Copy the diagram in Example 1. Draw and name the third midsegment.
2. In Example 1, suppose the distance UW is 81 inches. Find VS.
ANSWER
81 in.
ANSWER
UW
Warm-Up ExercisesEXAMPLE 2 Use the Midsegment Theorem
In the kaleidoscope image, AE BE and AD CD . Show that CB DE .
SOLUTION
Because AE BE and AD CD , E is the midpoint of AB and D is the midpoint of AC by definition.
Then DE is a midsegment of ABC by definition and CB DE by the Midsegment Theorem.
Warm-Up ExercisesEXAMPLE 3 Place a figure in a coordinate plane
Place each figure in a coordinate plane in a way that is convenient for finding side lengths. Assign coordinates to each vertex.
a. A rectangle b. A scalene triangle
SOLUTION
It is easy to find lengths of horizontal and vertical segments and distances from (0, 0), so place one vertex at the origin and one or more sides on an axis.
Warm-Up ExercisesEXAMPLE 3 Place a figure in a coordinate plane
a. Let h represent the length and k represent the width.
b. Notice that you need to use three different variables.
Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3
3. In Example 2, if F is the midpoint of CB , what do you know about DF ?
ANSWER
DF AB and DF is half the length of AB.
4. Show another way to place the rectangle in part (a) of Example 3 that is convenient for finding side lengths. Assign new coordinates.
ANSWER
DF is a midsegment of ABC.
Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3
5. Is it possible to find any of the side lengths in part (b) of Example 3 without using the Distance Formula? Explain.
Yes; the length of one side is d.
ANSWER
6. A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex.
ANSWER
(m, m)
Warm-Up ExercisesEXAMPLE 4 Apply variable coordinates
SOLUTION
Place PQO with the right angle at the origin. Let the length of the legs be k. Then the vertices are located at P(0, k), Q(k, 0), and O(0, 0).
Place an isosceles right triangle in a coordinate plane. Then find the length of the hypotenuse and the coordinates of its midpoint M.
Warm-Up ExercisesEXAMPLE 4 Apply variable coordinates
Use the Distance Formula to find PQ.
PQ = (k – 0) + (0 – k)22 = k + (– k)
2 2 = k + k2 2
= 2k 2 = k 2
Use the Midpoint Formula to find the midpoint M of the hypotenuse.
M( )0 + k , k + 02 2 = M( , )k
2k2
Warm-Up ExercisesEXAMPLE 5 Prove the Midsegment Theorem
GIVEN : DE is a midsegment of OBC.
PROVE : DE OC and DE = OC12
Write a coordinate proof of the Midsegment Theorem for one midsegment.
SOLUTION
STEP 1 Place OBC and assign coordinates. Because you are finding midpoints, use 2p, 2q, and 2r. Then find the coordinates of D and E.
D( )2q + 0, 2r + 02 2
= D(q, r) E( )2q + 2p, 2r + 02 2
= E(q+p, r)
Warm-Up ExercisesEXAMPLE 5 Prove the Midsegment Theorem
STEP 2 Prove DE OC . The y-coordinates of D and E are the same, so DE has a slope of 0. OC is on the x-axis, so its slope is 0.
STEP 3 Prove DE = OC. Use the Ruler Postulate12
to find DE and OC .
DE =(q + p) – q = p OC = 2p – 0 = 2p
Because their slopes are the same, DE OC .
So, the length of DE is half the length of OC
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
7. In Example 5, find the coordinates of F, the midpoint of OC . Then show that EF OB .
(p, 0); slope of EF = = ,
slope of OB = = , the slopes of
EF and OB are both , making EF || OB.
r 0(q + p) p q
r
2r 02q 0 q
r
qr
ANSWER
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
8. Graph the points O(0, 0), H(m, n), and J(m, 0). Is OHJ a right triangle? Find the side lengths and the coordinates of the midpoint of each side.
ANSWER
yes; OJ = m, JH = n,
HO = m2 + n2,
OJ: ( , 0), JH: (m, ),
HO: ( , )
2m
2n
2m
2n
Sample:
Warm-Up ExercisesDaily Homework Quiz
Use the figure below for Exercises 1–4.
1. If UV = 13, find RT.
2. If ST = 20, find UW.
ANSWER 26
ANSWER 10
3. If the perimeter of RST = 68 inches, find the Perimeter of UVW.
ANSWER 34 in.
Warm-Up ExercisesDaily Homework Quiz
4. If VW = 2x – 4, and RS = 3x – 3, what is VW?
ANSWER 6
5. Place a rectangle in a coordinate plane so itsvertical side has length a and its horizontal sidehas width 2a. Label the coordinates of eachvertex.
ANSWER
• You will use properties of midsegments and write coordinate proofs.
Essential Question: • How do you write a
coordinate proof?
• Assign coordinates to verticesthat are convenient for findinglengths.• Use coordinates to find midpoints,distances, and slopes.
Assign convenient coordinates tovertices and use the midpoint,distance, and slope formulas togenerate the proof.